MinT - Best Known (165−39, 165, s)-Nets in Base 4

Best Known (165−39, 165, s)-Nets in Base 4

(165−39, 165, 1036)-Net over F₄ — Constructive and digital

Digital (126, 165, 1036)-net over F₄, using

4¹ times duplication [i] based on digital (125, 164, 1036)-net over F₄, using
- trace code for nets [i] based on digital (2, 41, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(165−39, 165, 2079)-Net over F₄ — Digital

Digital (126, 165, 2079)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−39, 165, 415725)-Net in Base 4 — Upper bound on s

There is no (126, 165, 415726)-net in base 4, because

1 times m-reduction [i] would yield (126, 164, 415726)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 827189 636816 643193 672823 373391 088813 019288 927912 652221 855737 528609 054594 828295 002914 508820 799324 > 4¹⁶⁴ [i]